The bubble looks like in the image below.
It will only show in the standard (TouchWiz) launcher -- or in Apex Launcher, Nova Launcher, Smart Launcher 2.
It will not show in e.g. GO Launcher.
The functions to push the counter are launcher specific, and in the case of manufacturer launchers, are not documented. I didn't have any trouble with it on my Galaxy S4, though. The screenshot is from a Note 3.
Now, to turn this off -- it's in settings under "message notifications", and the setting is labeled:
Startseitensymbol Zähler
Bei bestimmten Startseiten und Hersteller